On a Problem of Statistical Inference in Null Recurrent Diffusions

We consider a particular example of statistical inference in null recurrent one-dimensional diffusions. In a first parametric model, we prove local asymptotic mixed normality (LAMN) and efficiency of the sequence of maximum likelihood estimates (MLE): its speed of convergence is n ^/2 with ranging over (0, 1). In a second semiparametric model (where in addition an unknown nuisance function with known compact support is included in the drift), we prove a local asymptotic minimax bound and specify asymptotically efficient estimates for the unknown parameter.